291 research outputs found
Hierarchical matrix arithmetic with accumulated updates
Hierarchical matrices can be used to construct efficient preconditioners for
partial differential and integral equations by taking advantage of low-rank
structures in triangular factorizations and inverses of the corresponding
stiffness matrices.
The setup phase of these preconditioners relies heavily on low-rank updates
that are responsible for a large part of the algorithm's total run-time,
particularly for matrices resulting from three-dimensional problems.
This article presents a new algorithm that significantly reduces the number
of low-rank updates and can reduce the setup time by 50 percent or more
Adaptive compression of large vectors
Numerical algorithms for elliptic partial differential equations frequently
employ error estimators and adaptive mesh refinement strategies in order to
reduce the computational cost.
We can extend these techniques to general vectors by splitting the vectors
into a hierarchically organized partition of subsets and using appropriate
bases to represent the corresponding parts of the vectors. This leads to the
concept of \emph{hierarchical vectors}.
A hierarchical vector with subsets and bases of rank requires
units of storage, and typical operations like the evaluation of norms and inner
products or linear updates can be carried out in
operations.
Using an auxiliary basis, the product of a hierarchical vector and an
-matrix can also be computed in operations,
and if the result admits an approximation with subsets in the
original basis, this approximation can be obtained in
operations. Since it is possible to compute
the corresponding approximation error exactly, sophisticated error control
strategies can be used to ensure the optimal compression.
Possible applications of hierarchical vectors include the approximation of
eigenvectors and the solution of time-dependent problems with moving local
irregularities
Complexity estimates for triangular hierarchical matrix algorithms
Triangular factorizations are an important tool for solving integral
equations and partial differential equations with hierarchical matrices
(-matrices).
Experiments show that using an -matrix LR factorization to solve
a system of linear questions is superior to direct inversion both with respect
to accuracy and efficiency, but so far theoretical estimates quantifying these
advantages were missing.
Due to a lack of symmetry in -matrix algorithms, we cannot hope
to prove that the LR factorization takes one third of the operations of the
inversion or the matrix multiplication, as in standard linear algebra. We can,
however, prove that the LR factorization together with two other operations of
similar complexity, i.e., the inversion and multiplication of triangular
matrices, requires not more operations than the matrix multiplication.
We can complete the estimates by proving an improved upper bound for the
complexity of the matrix multiplication, designed for recently introduced
variants of classical -matrices
Hybrid matrix compression for high-frequency problems
Boundary element methods for the Helmholtz equation lead to large dense
matrices that can only be handled if efficient compression techniques are used.
Directional compression techniques can reach good compression rates even for
high-frequency problems.
Currently there are two approaches to directional compression: analytic
methods approximate the kernel function, while algebraic methods approximate
submatrices. Analytic methods are quite fast and proven to be robust, while
algebraic methods yield significantly better compression rates.
We present a hybrid method that combines the speed and reliability of
analytic methods with the good compression rates of algebraic methods
Approximation of integral operators by Green quadrature and nested cross approximation
We present a fast algorithm that constructs a data-sparse approximation of
matrices arising in the context of integral equation methods for elliptic
partial differential equations.
The new algorithm uses Green's representation formula in combination with
quadrature to obtain a first approximation of the kernel function and then
applies nested cross approximation to obtain a more efficient representation.
The resulting -matrix representation requires units of storage for an matrix, where depends on the
prescribed accuracy
Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates
Many matrices appearing in numerical methods for partial differential
equations and integral equations are rank-structured, i.e., they contain
submatrices that can be approximated by matrices of low rank. A relatively
general class of rank-structured matrices are -matrices: they
can reach the optimal order of complexity, but are still general enough for a
large number of practical applications. We consider algorithms for performing
algebraic operations with -matrices, i.e., for approximating the
matrix product, inverse or factorizations in almost linear complexity. The new
approach is based on local low-rank updates that can be performed in linear
complexity. These updates can be combined with a recursive procedure to
approximate the product of two -matrices, and these products can
be used to approximate the matrix inverse and the LR or Cholesky factorization.
Numerical experiments indicate that the new method leads to preconditioners
that require units of storage, can be evaluated in
operations, and take operations to set
up
GCA- matrix compression for electrostatic simulations
We consider a compression method for boundary element matrices arising in the
context of the computation of electrostatic fields. Green cross approximation
combines an analytic approximation of the kernel function based on Green's
representation formula and quadrature with an algebraic cross approximation
scheme in order to obtain both the robustness of analytic methods and the
efficiency of algebraic ones. One particularly attractive property of the new
method is that it is well-suited for acceleration via general-purpose graphics
processors (GPUs)
Approximation of boundary element matrices using GPGPUs and nested cross approximation
The efficiency of boundary element methods depends crucially on the time
required for setting up the stiffness matrix. The far-field part of the matrix
can be approximated by compression schemes like the fast multipole method or
-matrix techniques. The near-field part is typically approximated
by special quadrature rules like the Sauter-Schwab technique that can handle
the singular integrals appearing in the diagonal and near-diagonal matrix
elements.
Since computing one element of the matrix requires only a small amount of
data but a fairly large number of operations, we propose to use general-purpose
graphics processing units (GPGPUs) to handle vectorizable portions of the
computation: near-field computations are ideally suited for vectorization and
can therefore be handled very well by GPGPUs. Modern far-field compression
schemes can be split into a small adaptive portion that exhibits divergent
control flows, and should therefore be handled by the CPU, and a vectorizable
portion that can again be sent to GPGPUs.
We propose a hybrid algorithm that splits the computation into tasks for CPUs
and GPGPUs. Our method presented in this article is able to reduce the setup
time of boundary integral operators by a significant factor of 19-30 for both
the Laplace and the Helmholtz equation in 3D when using two consumer GPGPUs
compared to a quad-core CPU
Approximation of the high-frequency Helmholtz kernel by nested directional interpolation
We present and analyze an approximation scheme for a class of highly
oscillatory kernel functions, taking the 2D and 3D Helmholtz kernels as
examples. The scheme is based on polynomial interpolation combined with
suitable pre- and postmultiplication by plane waves. It is shown to converge
exponentially in the polynomial degree and supports multilevel approximation
techniques. Our convergence analysis may be employed to establish exponential
convergence of certain classes of fast methods for discretizations of the
Helmholtz integral operator that feature polylogarithmic-linear complexity
An analysis of a butterfly algorithm
Butterfly algorithms are an effective multilevel technique to compress
discretizations of integral operators with highly oscillatory kernel functions.
The particular version of the butterfly algorithm considered here realizes the
transfer between levels by Chebyshev interpolation. We present a refinement of
the analysis that improves the stability estimates underlying the error bounds
- …